Probability density function chaotic flow

Dye structures of passive tracers placed in time-periodic chaotic flows evolve in an iterative fashion an entire structure is mapped into a new structure with persistent large-scale features, but finer and finer scale features are revealed at each period of the flow. After a few periods, strategically placed blobs of passive tracer reveal patterns that serve as templates for subsequent stretching and folding. Repeated action by the flow generates a lamellar structure consisting of stretched and folded striations, with thicknesses s(r), characterized by a probability density function, f(s,t), whose... 

The time invariance in the spatial distribution of p is another manifestation of the self-similarity of the structures generated by time- and spatially periodic chaotic flows. Such invariant statistical properties can be demonstrated by computing the probability density H(log p). If the probability density function of the scaled variable (p/(p is computed according to eq. (3-13), the distributions are scaled automatically by the mean density. 

Figure 3-30 The probability density function of the product concentration [PDF(cp)] in the tank is calculated from the experimental images after t = 1, 2, and 3 min. The experimental distributions in the 3D chaotic flow are invariant after scaling and match the concentration distribution of the model system. It is shown by overlaying PDF(cp) in the tank and in the sine flow at n = 3.

---